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I'm starting to learn the minimal model program. It seems there are two definitions for a variety $X$ with only terminal singularities to be minimal:

  1. $K_X$ is nef.
  2. Every birational morphism from $X$ to $Y$ must be an isomorphism, where $Y$ is another variety with only terminal singularities.

Suppose $X$ is a smooth minimal model of dimension greater than 2. Why are 1 and 2 equivalent definitions? In particular, why does 2 imply 1? Does the exceptional divisor contain an extremal ray?

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Thanks! This clarifies my understanding of exceptional divisor. By the way, if both $X$ and $Y$ are varieties of the same dimension, $f$ is a nontrivial birational morphism from $X$ to $Y$, does $X$ necessarily contain an exceptional divisor which is contracted by $f$? I know this is true when both $X$ and $Y$ are smooth. – user17314 Aug 25 '11 at 4:12
up vote 4 down vote accepted

1 implies 2 follows from the so called "negativity lemma", see for example Lemma 3.39 in the book by Kollar-Mori. The main point is that an effective linear combination of exceptional divisors can never be $f$-nef for a birational morphism $f:X \to Y$.

2 implies 1 is not true: consider $X = \mathbb{P}^n$.

(The condition 2 is the classical definition of minimal but Mori realised that 1 is a better definition.)

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